What is a sphere ?

If 'C' is a fixed point in space and 'a' is a non-negative real number, then the set of all points 'P' in space, such that the distance CP = a, is called a sphere.
The point 'C' is it's center and 'a' it's radius. A sphere with a = 0 (i.e, zero radius) is called point sphere.

Another definition : The locus of a variable point in space which moves such that its distance from a fixed point C in space is constant 'a' (&GreaterEqual; 0) is called a sphere.

We shall first discuss Vector and Cartesian equations of a sphere. The concept of scalar product in vector algebra is used to derive the vector equation. We shall use bold letters to represent a vector.

Vector form of equation of a sphere

The vector equation of the sphere with center 'C', whose position vector is c, and radius 'a' is given by
| r – c | = a
or r2 – 2r.c + c2 = a2

Conversely, if P(r) is any point satisfying
| r – c | = a, then CP = a and P lies on the sphere.

If OC = a (i.e, the origin of reference lies on the sphere), the equation of the sphere is r2 – 2r.c = 0
And if c = 0 (i.e, the origin of reference is the center of the sphere), the equation of the sphere is r2 = a2 or | r | = a

If A(a) and B(b) are the two end points of a diameter of the sphere, then its vector equation is
(r – a).(r – b) = 0
or r2 – r.(a + b) + a.b = 0

It is more common to denote the radius of a sphere by 'r' (and not 'a'). We now use the symbol Σ to represent a sphere with the center 'C' and radius 'r'.
Then in set-builder form
Σ = { P &Element; R3 : CP = r}
If r = 0, Σ is called a point sphere.
The equation of a sphere whose center is A(x0, y0, z0) and radius 'r' is

Diameter of a sphere :

If A and B are two points on Σ and if the line passes through its center, then is called as diameter line. The line sequent AB is called diameter of Σ.
The equation of a sphere on the join of A(x1, y1, z1) and B(x2, y2, z2) as diameter is

A diameter divides a sphere into two equal parts. Each is called a hemisphere.
If P(x0, y0, z0) is a point on the sphere and is distinct from A and B, then
(x0 – x1)(x0 – x2) + (y0 – y1)(y0 – y2) + (z0 – z1)(z0 – z2) = 0
We know (x0 – x1, y0 – y1, z0 – z1) and (x0 – x2, y0 – y2, z0 – z2)
are triads of direction numbers and respectively. This implies is perpendicular to .Therefore the angle in a hemisphere is right angle(90°).

The equation of a sphere in R3 is expressed of the form
x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0
where u2 + v2 + w2 &GreaterEqual; d
The above form is called the general form of the equation of a sphere.The center of above sphere is (– u, – v, – w)
It's radius is given by √(u2 + v2 + w2 – d)